Open loop cyclostationarity based timing recovery for accelerated timing acquisition in frequency selective channels

ABSTRACT

An error between the rate f sym  at which data are received and the rate f s  at which the data are sampled in is determined by processing a received signal with a nonlinear operator, performing a DFT on the processed signal to produce a plurality of DFT bins each characterized by a respective frequency, determining a dominant spectral component k 0  from at least two of the DFT bins whose frequencies are substantially close to the frequency of the dominant spectral component k 0 , and determining the data rate f sym  from the dominant spectral component k 0 .

RELATED APPLICATIONS

The present application claims the benefit of Provisional ApplicationSer. No. 60/336,071 filed on Oct. 25, 2001.

TECHNICAL FIELD OF THE INVENTION

The present invention relates to timing recovery in digital receivers.

BACKGROUND OF THE INVENTION

Digital receivers typically sample the received signal, and processthese samples in order to perform a variety of functions. It isimportant in such receivers to sample the received signal at theinstants when a data element such as a symbol is present. Accordingly,receivers recover the timing at which data elements are received and usethis timing to control sampling of the received signal.

Timing recovery deals with finding the ideal sample times to recover thereceived symbols from a received signal. Mismatch between thetransmitter and receiver clocks complicates the recovery process becauseslight differences between these clocks causes a signal to be sampled atthe wrong times, contributing to intersymbol interference.

One popular technique for recovering timing in a receiver is known asthe Gardner technique. FIG. 1 illustrates a waveform representing datathat might be received at a receiver. In the limited case of FIG. 1, thedata is +1 −1 −1 +1. This waveform assumes that the transmitted data isconfined to a binary constellation containing only values of −1 and +1,although it can be shown that the Gardner technique works as well forlarger constellations. As shown by FIG. 1, the received signal issampled at two times the data rate.

The Gardner technique essentially forms a timing error according tob(a−c). As long as the sampling is exactly synchronized with respect tothe received data, the quantity b(a−c) is zero. For example, in the caseof a transition from +1 to −1 as shown in FIG. 1, b is zero so that thequantity b(a−c) is zero. In the case of no transition so that twoadjacent bits have the same value such as −1 and −1 as shown in FIG. 2,b is no longer zero. However, a and c have the same value so that thequantity b(a−c) is again zero.

If the sampling timing is not exactly synchronized to the received data,the quantity b(a−c) is not zero but, instead, takes on a value that is ameasure of how far off the sampling is from the optimal samplinginstants. This value is the timing error and may be used tore-synchronize the sampler to the received data.

Although the Gardner technique works reasonably well in the case ofwhite Gaussian noise such as electronic thermal noise or atmosphericnoise, it does not work as well in the case of multi-path reception ofthe transmitted signal.

The present invention relates to the accurate recovery of timing even inthe presence of multi-path reception of a transmitted signal.

SUMMARY OF THE INVENTION

In accordance with one aspect of the present invention, a method isprovided to determine an error between the rate f_(sym) at which dataare received and the rate f_(s) at which the data are sampled in areceiver. The method comprises the following: processing a receivedsignal by a nonlinear operator; performing a DFT on the processed signalto produce a plurality of DFT bins each characterized by a respectivefrequency; determining a dominant spectral component k₀ from at leasttwo of the DFT bins whose frequencies are substantially close to thefrequency of the dominant spectral component k₀; and, determining thedata rate f_(sym) from the dominant spectral component k₀.

In accordance with another aspect of the present invention, a method isprovided to correct the sampling rate f_(s) of a receiver so that thesampling rate f_(s) matches the rate f_(sym) at which data are receivedby the receiver. The method comprises the following: processing areceived signal by a nonlinear operator; performing a DFT on theprocessed signal to produce a plurality of DFT bins each characterizedby a respective frequency; determining a dominant spectral component k₀from two of the DFT bins whose frequencies are substantially close tothe frequency of the dominant spectral component k₀; determining anerror based on the dominant spectral component k₀, wherein the errorindicates an offset between f_(s) and f_(sym); and, adjusting thesampling rate f_(s) to reduce the error.

In accordance with still another aspect of the present invention, amethod is provided to determine a relationship between a sampling ratef_(s) at which a receiver samples received data and a data receive ratef_(sym) at which the data are received by the receiver. The methodcomprises the following: processing a received signal by a nonlinearoperator; sampling the processed received signal to produce signalsamples; performing a DFT on the signal samples to produce a pluralityof DFT bins each characterized by a corresponding frequency; determininga dominant spectral component k₀ from two of the DFT bins whosecorresponding frequencies are equal or closest to f_(sym); and,determining the relationship between the sampling rate f_(s) and thedata receive rate f_(sym) based on the dominant spectral component k₀.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features and advantages will become more apparent from adetailed consideration of the invention when taken in conjunction withthe drawings in which:

FIGS. 1 and 2 illustrate waveforms useful in explaining a prior artapproach to the correction of the sampling rate in a receiver;

FIG. 3 illustrates an 8 VSB spectrum at baseband;

FIG. 4 illustrates a timing error detector according to an embodiment ofthe present invention;

FIG. 5 illustrates a DFT (Discrete Fourier Transform) of a receivedsignal;

FIG. 6 illustrates a DFT of a squared received signal;

FIG. 7 illustrates a DFT of a received signal raised to the fourthpower;

FIG. 8 illustrates an arrangement for correcting the sampling frequencyof a receiver based on the timing error detected by the timing errordetector shown in FIG. 4; and,

FIG. 9 illustrates a loop filter that can be used on the arrangement ofFIG. 8.

DETAILED DESCRIPTION

While the timing estimator disclosed herein is equally applicable toother linear modulation techniques, the present invention is disclosedherein with particular reference to 8 VSB modulation, such as thatcurrently employed in digital television. The 8 VSB signal is a linearlymodulated 8-ary PAM signal with real-valued symbols b[k] and a complexpulse shape p_(v)(t). The transmitted signal may be expressed by thefollowing equation:

$\begin{matrix}{{x_{v}(t)} = {\sum\limits_{k}{{b\lbrack k\rbrack}{{p_{v}\left( {t - {kT}_{sym}} \right)}.}}}} & (1)\end{matrix}$The symbols sent every T_(sym) seconds are assumed to be independent andidentically distributed. These symbols are taken from the alphabethaving the following amplitudes:b[m]=(2m−9)d, for m=1,2, . . . , 8where 2d is the distance between adjacent symbols. The 8 VSB pulse shapep_(v)(t) is the complex root raised cosine pulse shape whose spectrum isshown in FIG. 3.

The 8 VSB signal x_(v)(t) is passed through a channel denoted c(t) andis received at the receiver to yield a noisy signal according to thefollowing equation:r _(v)(t)=c(t)*x _(v)(t)+ν(t)  (2)where * denotes convolution and ν(t) is the zero-mean complex additivewhite Gaussian noise with independent and identically distributed realand imaginary components, each with a variance σ_(n) ². In “all-digital”receivers, the received signal is, prior to matched filtering,oversampled by some multiple (such as two) of the nominal symbol ratef_(sym)=(1/T_(sym)) to give the digital signal r_(v)[n].

The cyclostationarity technique disclosed herein recovers informationabout the symbol rate by first passing the received signal through anon-linearity to detect a discrete spectral component corresponding tothe symbol rate and then filtering or using some other technique todetermine the frequency of the spectral component.

In one embodiment of the present invention, this non-linearity issquaring. Accordingly, the received signal represented by equation (2)is squared to produce the following equation:

$\begin{matrix}\begin{matrix}{{r_{v}^{2}(t)} = \left( {{{x_{v}(t)}*{c(t)}} + {\upsilon(t)}} \right)^{2}} \\{= {\left( {{x_{v}(t)}*{c(t)}} \right)^{2} + {2\;{\upsilon(t)}\left( {{x_{v}(t)}*{c(t)}} \right)} + {v^{2}(t)}}}\end{matrix} & (3)\end{matrix}$An interesting result of simply squaring the signal rather than squaringthe magnitude of the signal is that the complex-valued noise averages tozero. This result means that the expected value of the noise, ε{v²(t)},is zero. Moreover, because the noise is assumed zero-mean andindependent of the data, the expected value of the cross terms inequation (3) is also zero. Therefore, taking the expected value removesall terms involving the noise v(t) according to the following equation:ε{r _(v) ²(t)}=ε{(x _(v)(t)*c(t))²}  (4)Combining equations (1) and (4) produces the following equation:

$\begin{matrix}{{ɛ\left\{ {r_{v}^{2}(t)} \right\}} = {\sigma_{b}^{2}{\sum\limits_{k}{p_{c}^{2}\left( {t - {kT}_{sym}} \right)}}}} & (5)\end{matrix}$where p_(c)(t)=p_(v)(t)*c(t)

Equation (5) describes a signal that is periodic with a period T_(sym).Therefore, equation (5) can be expressed as a Fourier series accordingto the following equation:

$\begin{matrix}{{ɛ\left\{ {r_{v}^{2}(t)} \right\}} = {\sum\limits_{k}{c_{k}{\mathbb{e}}^{{j2}\;\pi\;{kf}_{sym}}}}} & (6)\end{matrix}$where c_(k) are the Fourier coefficients.

For a band limited signal, squaring the signal doubles the bandwidth ofthe signal. For the baseband 8 VSB signal, the baseband spectrum extendsfrom −0.31 MHz to 5.69 MHz. Thus, the squared signal has energy in thefrequency interval from −0.62 MHz to 11.38 MHz. Because f_(sym)=10.72MHz for the 8 VSB signal, all Fourier series coefficients are zeroexcept c₀ and c₁.

While the relationship between the receiver's sampling rate f_(s) andthe true symbol rate f_(sym) is not known precisely at the receiver, itis reasonable to assume that the receiver has a good initial estimate ofthe true symbol rate. Use of a crystal oscillator having an accuracy of±100 parts per million (ppm) means that it can reasonably be assumedthat the receiver can determine the symbol rate f_(sym) of the receivedsignal relative to its own clock within ±100 ppm. Based on theseassumptions, if a data signal has a symbol rate of 10 MHZ, the symbolrate is known within ±1000 Hz at the receiver. However, even this smalloffset can cause severe error performance over time unless this error iscorrected.

In one embodiment of the present invention, the received signal isoversampled to produce samples r_(v)[n]. As shown in FIG. 4, theoversampled signal r_(v)[n] is supplied to a timing error detector 10that includes a bandpass filter 12 operating at half the nominal symbolrate (f_(sym)/2). The bandpass filter 12 bandpass filters theoversampled signal r_(v)[n]. The impulse response of the bandpass filter12, for example, may be given by the following equation:

$\begin{matrix}{{h_{bpf}\lbrack n\rbrack} = {{\cos\left( {\pi\frac{n}{2\; K}} \right)}{\cos\left( {\pi\frac{n}{U_{os}}} \right)}}} & (7)\end{matrix}$where K is the filter length and U_(os) is the oversampling factor.

The resulting signal r_(bpf)[n] is passed through a non-linear operator14 that imposes a non-linear operation, such as squaring, on the signalr_(bpf)[n] in order to boost the spectral line at the digital frequencycorresponding to the symbol rate f_(sym) relative to the other spectrallines. For example, FIG. 5 shows a DFT of a received signal without thenon-linear operation. The symbol rate f_(sym) is buried among the otherfrequency components of the received signal and cannot be readilydistinguished. FIG. 6 shows the effects on the DFT of squaring thereceived signal samples. The symbol rate f_(sym) rises above the otherfrequency components and, thereby, is more readily distinguished. Asalso can be seen by comparing FIGS. 5 and 6, the bandwidth has doubledwhen the samples of the received signal are squared.

The non-linearity is preferably chosen with the nominal sampling rate inmind so that the signal's spectrum is not aliased near the symbol ratef_(sym). For an 8 VSB signal that is squared, sampling at twice thenominal symbol rate is enough to prevent aliasing of the squared signal.

The signal x[n] resulting from the non-linear operator 14 is modeledover a small frequency interval around the true symbol rate as a singlecomplex exponential plus white noise according to the followingequation:x[n]=αz[n]+v[n]  (7)where

$\begin{matrix}{{z\lbrack n\rbrack} = {\mathbb{e}}^{j\frac{2\;\pi\; k_{0}n}{N}}} & (8)\end{matrix}$where α is a complex scalar, and where v[n] is the noise component. Itis then possible to compute two Discrete Fourier Transform (DFT) values{tilde over (Z)}(k) and {tilde over (Z)}(k+1) near the nominal value ofk₀ that corresponds to the rate f_(sym) at which data is received, todetermine k₀ from these values, and to determine the relationshipbetween f_(sym) and f_(s) from k₀. Subspace based averaging may be usedto determine these values such that the effect of noise is minimized.

Accordingly, a timing error estimator 16 performs a DFT on x[n]. Aconjugate centrosymmetrized version of the kth bin of the resultingN-point DFT is given by the following equation:

$\begin{matrix}{{\overset{\sim}{X}(k)} = {\sum\limits_{n = 0}^{N - 1}{{x\lbrack n\rbrack}{\mathbb{e}}^{{- j}\frac{2\;\pi\; k}{N}{({n - \frac{N - 1}{2}})}}}}} & (9)\end{matrix}$where x[n] are the received signal samples. Equation (9) can bere-written as a product of vectors according to the following equation:{tilde over (X)}(k)={tilde over (w)} _(k) ^(H) x  (10)where

$\begin{matrix}{{\overset{\sim}{w}}_{k}^{H} = {\left\lbrack {{\mathbb{e}}^{j\frac{2\;\pi\; k}{N}{(\frac{N - 1}{2})}}{\mathbb{e}}^{j\frac{2\;\pi\; k}{N}{(\frac{N - 3}{2})}}\cdots\mspace{11mu}{\mathbb{e}}^{{- j}\frac{2\;\pi\; k}{N}{(\frac{N - 1}{2})}}} \right\rbrack \in C^{N}}} & (11)\end{matrix}$and wherex=[x[0]x[1] . . . x[N−1]]^(T) εC ^(N)  (12)C^(N) represents the N dimensional complex vector space comprisingcomplex vectors each of length N. The rotated DFT vector {tilde over(w)}_(k) is termed conjugate centrosymmetric because its mth element isequal to the conjugate of its (N-m)th element for m=0, . . . , N−1.{tilde over (X)}(k) may be found by first finding the DFT value X(k) andby then post multiplying this value by

${\mathbb{e}}^{j\frac{2\;\pi\; k}{N}{(\frac{N - 1}{2})}}.$

If z[n] is the single complex exponential in equation (8), then the kthbin of the conjugate centrosymmetrized DFT of z[n] evaluated at DFT bink is given by the following equation:

$\begin{matrix}{{\overset{\sim}{Z}(k)} = {\frac{\sin\;\left( {\pi\left( {k - k_{0}} \right)} \right)}{\sin\left( {\frac{\pi}{N}\left( {k - k_{0}} \right)} \right)}{\mathbb{e}}^{j\frac{2\;\pi\; k_{0}}{N}{(\frac{N - 1}{2})}}}} & (13)\end{matrix}$and the conjugate centrosymmetrized DFT of z[n] evaluated at DFT bin k+1is given by the following equation:

$\begin{matrix}{{\overset{\sim}{Z}\left( {k + 1} \right)} = {\frac{\sin\left( {\pi\left( {k + 1 - k_{0}} \right)} \right)}{\sin\left( {\frac{\pi}{N}\left( {k + 1 - k_{0}} \right)} \right)}{\mathbb{e}}^{j\frac{2\;\pi\; k_{0}}{N}{(\frac{N - 1}{2})}}}} & (14)\end{matrix}$Using the trigonometric identity sin(A+B)=sin(A)cos(B)+cos(A)sin(B), thenumerator of equation (14) becomes −sin(π(k−k₀)). Therefore, dividingequation (13) by equation (14) yields the following equation:

$\begin{matrix}{\frac{\overset{\sim}{Z}(k)}{\overset{\sim}{Z}\left( {k + 1} \right)} = {- \frac{\sin\left( {\frac{\pi}{N}\left( {k + 1 - k_{0}} \right)} \right)}{\sin\left( {\frac{\pi}{N}\left( {k - k_{0}} \right)} \right)}}} & (15)\end{matrix}$Applying the trigonometric identity given above to equation (15)produces the following equation:

$\begin{matrix}{k_{0} = {\frac{N}{\pi}{\tan^{- 1}\left( \frac{{{\overset{\sim}{Z}(k)}{\sin\left( {\frac{\pi}{N}k} \right)}} + {{\overset{\sim}{Z}\left( {k + 1} \right)}{\sin\left( {\frac{\pi}{N}\left( {k + 1} \right)} \right)}}}{{{\overset{\sim}{Z}(k)}{\cos\left( {\frac{\pi}{N}k} \right)}} + {{\overset{\sim}{Z}\left( {k + 1} \right)}{\cos\left( {\frac{\pi}{N}\left( {k + 1} \right)} \right)}}} \right)}}} & (16)\end{matrix}$Based on k₀, the true symbol rate, i.e., the symbol rate of the receiveddata, can be determined from the following equation:

$\begin{matrix}{f_{sym} = {\frac{k_{0}}{N}f_{s}}} & (17)\end{matrix}$where N is the number of bins in the DFT and f_(s) is the samplingfrequency of the receiver.

The relationship between f_(sym) and f_(s) should ideally be the designrelationship. For example, if the sampling rate is nominally set attwice the symbol rate used by the transmitter, then ideallyf_(sym)=f_(s)/2. If the actual symbol rate of the received data, asdetermined by equations (16) and (17), and f_(s) do not have this idealrelationship, then the sampling rate of the receiver is adjusted untilthe ideal relationship is achieved.

The timing error estimator 16 can also determine the timing error bysubtracting k₀ from k. This timing error may be used to appropriatelyadjust the sampling frequency of the receiver in order to correctlysample the received signal.

As discussed above, the non-linear operator 14 may be arranged to squarethe signal from the bandpass filter 12. Instead, other non-linearitiescan be imposed on the received signal. For example, the received signalmay be raised to the fourth power, i.e., (r_(v)[n])⁴. FIG. 7 shows theeffects on the DFT of raising the received signal samples to the fourthpower. The symbol rate f_(sym) rises even more above the other frequencycomponents and, thereby, is even more readily distinguished than is thecase of simply squaring the samples. As also can be seen by comparingFIGS. 5 and 7, the bandwidth has quadrupled when the samples of thereceived signal are raised to the fourth power. For an 8 VSB signal thatis raised to the fourth power, sampling at four times the nominal symbolrate is enough to prevent aliasing of the signal raised to the fourthpower. When the non-linear operator 14 raises the received signalsamples to the fourth power, the bandpass filter 12 becomes unnecessary.

It is preferable for the process described above that values for N and kbe chosen so that k<k₀<k+1 and so that N<fs/(2Δf) where Δf is thetolerance of the a priori estimate. For example, in the case where thesamples are raised to the fourth power so that the sampling is at aboutfour times the symbol rate, N can be set at 4096, and k can be set at1024 representing the nominal symbol rate f_(sym). The frequencyassociated with bin k+1 is the next higher frequency. The bin k−1 can beused instead of the bin k+1.

FIG. 8 shows an arrangement for correcting the sampling frequency of thereceiver. The received signal is re-sampled by a re-sampler 22 thatreceives the samples from an upstream sampler, that effectivelyreconstructs an analog signal from the samples, and that re-samples thereconstructed analog signal in response to a sampling frequency suppliedby a numerically controlled oscillator 24. The samples at the output ofthe re-sampler 22 are processed by a timing error estimator 26 to detectthe error between f_(s) and f_(sym) as described above, and this erroris filtered by a loop filter 28. The filtered error from the loop filter28 then controls the numerically controlled oscillator 24 to produce thecorrect sampling frequency for the re-sampler 22. A matched filter maybe located in the output of the re-sampler 22 upstream of the timingerror estimator 26, or the matched filter may instead be locatedupstream of the re-sampler 22.

FIG. 9 shows an example of a loop filter 40 that can be used for theloop filter 28 of FIG. 8. The loop filter 40 is a low pass filter thatsmoothes the error provided by the timing error estimator 26. Thesmoothed error is denoted in FIG. 9 as l[n]. The quantities α₁ and α₂are constants.

If the noise component v[n] in x[n] is removed, {tilde over (X)}(k,n)and {tilde over (X)}(k+l,n) can be used in equation (16) in place of{tilde over (Z)}(k) and {tilde over (Z)}(k+1) because multiplication bya scalar has no effect on equation (16). One way of removing the noisecomponent in x[n] is to perform subspace-based, or eigenvalue-based,averaging in the frequency domain.

Let {tilde over (X)}(k,n) and {tilde over (X)}(k+1,n) be conjugatecentrosymmetric N point DFT values of {x[n−N+1], . . . , x[n]} atfrequency bins k and k+1, respectively. These DFT values can be groupedas a vector X=[{tilde over (X)}(k,n){tilde over (X)}(k+1,n)]^(T). Then,the real-valued auto-correlation matrix R_(xx) of the received signal issimply the following expected value: ε{X[n]X^(H)[n]. This matrix has thefollowing form:

${{Re}\left\{ {{\hat{R}}_{xx}\lbrack n\rbrack} \right\}} = \begin{bmatrix}a & b \\c & d\end{bmatrix}$The eigenvectors associated with this matrix have the following form:

$e_{i} = {{{\varsigma\begin{bmatrix}\frac{\lambda_{i} - d}{c} & 1\end{bmatrix}}^{T}\varsigma} \in C^{l}}$$e_{i} = {\xi\left\lbrack \begin{matrix}\frac{b}{\lambda_{1} - a} & {{\left. 1 \right\rbrack^{T}\xi} \in C^{l}}\end{matrix} \right.}$where λ_(l) are eigenvalues given by the following equation:

$\lambda_{i} = \frac{a + {d \pm \sqrt{\left( {a - d} \right)^{2} + {4{bc}}}}}{2}$The eigenvector associated with the largest eigenvalue of the real partof the sample correlation matrix {circumflex over (R)}_(xx)[n] is amultiple of Z=[{tilde over (Z)}(k) {tilde over (Z)}(k+1)]^(T).Therefore, these values of {tilde over (Z)}[k] and {tilde over (Z)}[k+1]can be derived from this eigenvector and used in equation (16) tocompute k₀ which can be used in equation (17) to determine therelationship between f_(s) and f_(sym).

Because R_(xx) is real valued, only the real part of the vector productX[n]X^(H)[n] need be included when forming the sample correlationmatrix. Because {circumflex over (R)}_(xx)[n] is a 2×2 matrix, a simpleclosed form expression is available for the eigenvectors.

The first DFT values {tilde over (X)}(k,N−1) and {tilde over(X)}(k+1,N−1) may be computed gradually (without the need to store orcompute all of the twiddle factors

${w_{N}^{nk} = {\mathbb{e}}^{{- j}\frac{2\;\pi\; k\; n}{N}}},$n=0, . . . , N−1) using the well known Goertzel algorithm. Subsequentvalues of {tilde over (X)}(k,n) and {tilde over (X)}(k+1,n) for n>N−1may be found recursively. If X(k,n) is the kth bin of the N point DFT ofthe current and past N−1 values of x[n], then X(k,n) is given by thefollowing equation:

$\begin{matrix}{{X\left( {k,n} \right)} = {\sum\limits_{m = 0}^{N - 1}{{x\left\lbrack {m + n - \left( {N - 1} \right)} \right\rbrack}{\mathbb{e}}^{{- j}\frac{2\pi\; k\; m}{N}}}}} & (18)\end{matrix}$and X(k,n) may be expressed in terms of X(k,n−1) according to thefollowing equation:

$\begin{matrix}\begin{matrix}{{X\left( {k,n} \right)} = {{\left( {{X\left( {k,{n - 1}} \right)} - {x\left\lbrack {n - N} \right\rbrack}} \right){\mathbb{e}}^{{- j}\;\frac{2\;\pi\; k}{N}}} +}} \\{{x\lbrack n\rbrack}{\mathbb{e}}^{{- j}\;\frac{2\;\pi\;{k{({N - 1})}}}{N}}}\end{matrix} & (19)\end{matrix}$for n>N−1. Use of equation (19) requires only two complexmultiplications and two additions, which is a significant computationalsavings over equation (18). Furthermore, when k is an integer, equation(19) simplifies even more and requires only one complex multiplicationand two additions according to the following equation:

${X\left\lbrack {k,n} \right\rbrack} = {\left( {{X\left\lbrack {k,{n - 1}} \right\rbrack} - {x\left\lbrack {n - N} \right\rbrack} + {x\lbrack n\rbrack}} \right){\mathbb{e}}^{j\frac{2\;\pi\; k}{N}}}$

Accordingly, when averaging is used as described above, as each newsample is received, the sample is passed through the non-linearoperation. Two conjugate centrosymmetric DFT values are found using thecurrent sample and the previous N−1 samples. These values are used toupdate the real part of the sample correlation matrix, and theeigenvector having the largest eigenvalue is extracted to give a scaledestimate of {tilde over (Z)}(k) and {tilde over (Z)}(k+1). These valuesare used as per equations (16) and (17) to determine the symbolfrequency of the received data.

Modifications of the present invention will occur to those practicing inthe art of the present invention. Accordingly, the description of thepresent invention is to be construed as illustrative only and is for thepurpose of teaching those skilled in the art the best mode of carryingout the invention. The details may be varied substantially withoutdeparting from the spirit of the invention, and the exclusive use of allmodifications which are within the scope of the appended claims isreserved.

1. A method for determining a rate f_(sym) at which data are receivedcomprising: processing a received signal by a nonlinear operator;performing a DFT on the processed signal to produce a plurality of DFTbins each characterized by a respective frequency; determining adominant spectral component k₀ from at least two of the DFT bins whosefrequencies are substantially close to the frequency of the dominantspectral component k₀; and, determining the data rate f_(sym) from thedominant spectral component k₀.
 2. The method of claim 1 wherein theprocessing of the received signal by a nonlinear operator comprisessquaring the received signal.
 3. The method of claim 2 wherein theprocessing of the received signal comprises sampling the received signalat about twice the rate at which the data are received.
 4. The method ofclaim 1 wherein the determining of a dominant spectral component k₀comprises determining the dominant spectral component k₀ according tothe following equation:$k_{0} = {\frac{N}{\pi}{\tan^{- 1}\left( \frac{{{\overset{\sim}{Z}(k)}{\sin\left( {\frac{\pi}{N}k} \right)}} + {{\overset{\sim}{Z}\left( {k + 1} \right)}{\sin\left( {\frac{\pi}{N}\left( {k + 1} \right)} \right)}}}{{{\overset{\sim}{Z}(k)}{\cos\left( {\frac{\pi}{N}k} \right)}} + {{\overset{\sim}{Z}\left( {k + 1} \right)}{\cos\left( {\frac{\pi}{N}\left( {k + 1} \right)} \right)}}} \right)}}$wherein N comprises the number of points in the DFT, k and k+1 comprisethe DFT bins whose frequencies are substantially close to the frequencyof the dominant spectral component k₀, and {tilde over (Z)}(k) and{tilde over (Z)}(k+1) represent the energy in the bins k and k+1.
 5. Themethod of claim 4 further comprising determining an error between thedata rate f_(sym) and a rate f_(s) at which the data are sampled in areceiver in accordance with a relationship between the sampling ratef_(s) and the data rate f_(sym) based on the dominant spectral componentk₀.
 6. The method of claim 5 wherein the relationship comprisesf_(sym)=(k₀/N)f_(s).
 7. The method of claim 6 wherein the processing ofthe received signal by a nonlinear operator comprises squaring thereceived signal.
 8. The method of claim 7 wherein the sampling of thereceived signal comprises sampling the received signal at about twicethe rate at which the data are received.
 9. The method of claim 1wherein the determining of a dominant spectral component k₀ of theprocessed received signal comprises: determining an autocorrelation ofthe received signal; selecting an eigenvector of the autocorrelationhaving the largest eigenvalue; and, determining the dominant spectralcomponent k₀ based on the selected eigenvector.
 10. The method of claim1 wherein the determining of a dominant spectral component k₀ of theprocessed received signal comprises determining the dominant spectralcomponent k₀ based on subspace based averaging.
 11. A method forcorrecting a sampling rate f_(s) of a receiver so that the sampling ratef_(s) matches a rate f_(sym) at which data are received by the receiver,the method comprising: processing a received signal by a nonlinearoperator; performing a DFT on the processed signal to produce aplurality of DFT bins each characterized by a respective frequency;determining a dominant spectral component k₀ from two of the DFT binswhose frequencies are substantially close to the frequency of thedominant spectral component k₀; determining an error based on thedominant spectral component k₀, wherein the error indicates an offsetbetween f_(s) and f_(sym); and, adjusting the sampling rate f_(s) toreduce the error.
 12. The method of claim 11 wherein the processing ofthe received signal by a nonlinear operator comprises squaring thereceived signal.
 13. The method of claim 12 wherein the processing ofthe received signal comprises sampling the received signal at abouttwice the rate at which the data are received.
 14. The method of claim11 wherein the processing of a received signal by a nonlinear operatorcomprises sampling the received signal at the sampling rate f_(s) andprocessing the received signal samples by the nonlinear operator, andwherein the determining of a dominant spectral component k₀ comprisesdetermining the dominant spectral component k₀ according to thefollowing equation:$k_{0} = {\frac{N}{\pi}{\tan^{- 1}\left( \frac{{{\overset{\sim}{Z}(k)}{\sin\left( {\frac{\pi}{N}k} \right)}} + {{\overset{\sim}{Z}\left( {k + 1} \right)}{\sin\left( {\frac{\pi}{N}\left( {k + 1} \right)} \right)}}}{{{\overset{\sim}{Z}(k)}{\cos\left( {\frac{\pi}{N}k} \right)}} + {{\overset{\sim}{Z}\left( {k + 1} \right)}{\cos\left( {\frac{\pi}{N}\left( {k + 1} \right)} \right)}}} \right)}}$wherein N comprises the number of points in the DFT, k and k+1 comprisethe DFT bins whose frequencies are substantially close to the frequencyof the dominant spectral component k₀, and {tilde over (Z)}(k) and{tilde over (Z)}(k+1) represent the energy in the bins k and k+1. 15.The method of claim 14 wherein the determining of the error comprisesdetermining a relationship between the sampling rate f_(s) and the datarate f_(sym) based on the dominant spectral component k₀.
 16. The methodof claim 15 wherein the relationship comprises f_(sym)=(k₀/N)f_(s). 17.The method of claim 14 wherein the processing of the received signal bya nonlinear operator comprises squaring the received signal.
 18. Themethod of claim 17 wherein the sampling of the received signal comprisessampling the received signal at about twice the rate at which the dataare received.
 19. The method of claim 11 wherein the determining of adominant spectral component k₀ of the processed received signalcomprises: determining an autocorrelation of the received signal;selecting an eigenvector of the autocorrelation having the largesteigenvalue; and, determining the dominant spectral component k₀ based onthe selected eigenvector.
 20. The method of claim 11 wherein thedetermining of a dominant spectral component k₀ of the processedreceived signal comprises determining the dominant spectral component k₀based on subspace based averaging.
 21. The method of claim 11 whereinthe adjusting of the sampling rate f_(s) to reduce the error comprises:loop filtering the error; adjusting the frequency of a numericallycontrolled oscillator; and, resampling the received signal based on theadjusted frequency.
 22. A method for determining a relationship betweena sampling rate f_(s) at which a receiver samples received data and adata receive rate f_(sym) at which the data are received by thereceiver, the method comprising: processing a received signal by anonlinear operator; sampling the processed received signal to producesignal samples; performing a DFT on the signal samples to produce aplurality of DFT bins each characterized by a corresponding frequency;determining a dominant spectral component k₀ from two of the DFT binswhose corresponding frequencies are equal or closest to f_(sym); and,determining the relationship between the sampling rate f_(s) and thedata receive rate f_(sym) based on the dominant spectral component k₀.23. The method of claim 22 wherein the relationship comprisesf_(sym)=(k₀/N)f_(s), and wherein N comprises the number of points in theDFT.
 24. The method of claim 23 wherein the determining of a dominantspectral component k₀ comprises evaluating the DFT at bin k, evaluatingthe DFT at bin k+1, and determining the dominant spectral component k₀based on the evaluations, and wherein k≦k₀≦k+1.
 25. The method of claim24 wherein the processing of the received signal by a nonlinear operatorcomprises squaring the received signal.
 26. The method of claim 25wherein f_(s)=2f_(sym).
 27. The method of claim 24 wherein theevaluating of the DFT at bin k and the evaluating the DFT at bin k+1comprises: determining an autocorrelation of the received signal;selecting an eigenvector of the autocorrelation having the largesteigenvalue; and, determining the dominant spectral component k₀ based onthe selected eigenvector.
 28. The method of claim 24 wherein theevaluating of the DFT at bin k and the evaluating the DFT at bin k+1comprises determining the dominant spectral component k₀ based onsubspace based averaging.
 29. The method of claim 22 wherein theprocessing of the received signal by a nonlinear operator comprisessquaring the received signal.
 30. The method of claim 29 whereinf_(s)=2f_(sym).